The average of 8, 13, x, and y is 6. The average of 15, 9, x and x is 8. What is the value of y?

Options

A. `-1`
B. `0`
C. `4`
D. `6`

Answer

To solve the problem step by step, we will use the information given about the averages to create equations and solve for the unknowns \(x\) and \(y\).

### Step 1: Set up the first equation from the first average

The average of the numbers \(8\), \(13\), \(x\), and \(y\) is given as \(6\). The formula for the average is:

\[ \text{Average} = \frac{\text{Sum of values}}{\text{Number of values}} \]

So, we can write:

\[ \frac{8 + 13 + x + y}{4} = 6 \]

### Step 2: Simplify the first equation

Multiply both sides by \(4\) to eliminate the denominator:

\[ 8 + 13 + x + y = 24 \]

Now, combine \(8\) and \(13\):

\[ 21 + x + y = 24 \]

Subtract \(21\) from both sides:

\[ x + y = 3 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation from the second average

The average of the numbers \(15\), \(9\), \(x\), and \(x\) is given as \(8\). We can write:

\[ \frac{15 + 9 + x + x}{4} = 8 \]

### Step 4: Simplify the second equation

Again, multiply both sides by \(4\):

\[ 15 + 9 + x + x = 32 \]

Combine \(15\) and \(9\):

\[ 24 + 2x = 32 \]

Subtract \(24\) from both sides:

\[ 2x = 8 \]

Now, divide by \(2\):

\[ x = 4 \quad \text{(Equation 2)} \]

### Step 5: Substitute \(x\) back into Equation 1

Now that we have \(x = 4\), we can substitute this value back into Equation 1:

\[ 4 + y = 3 \]

### Step 6: Solve for \(y\)

Subtract \(4\) from both sides:

\[ y = 3 - 4 \]

Thus,

\[ y = -1 \]

### Final Answer

The value of \(y\) is \(-1\).

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